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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习2.3-2.5}
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\date{2024 年 3 月 25 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
证明：点集 $F$ 为闭集的充分必要条件是 $F=\overline{F}$.

\vspace{0.2cm}

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\item  %Problem 02
证明：任意多个开集的并集仍是开集，有限多个开集的交集仍是开集。

\vspace{0.2cm}

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\item  %Problem 03
证明：任意多个闭集的交集仍是闭集，有限多个闭集的并集仍是闭集。
举例说明任意多个闭集的并集不一定是闭集。

\vspace{0.2cm}

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\item  %Problem 04
证明：$\mathbb{R}^n$ 中的有界闭集必定是紧集，紧集必定是有界闭集。

\vspace{0.2cm}

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\item  %Problem 05
什么时候称 $E\subseteq\mathbb{R}^n$ 是一个完备集？举出完备集的一些例子。

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\item  %Problem 06
证明直线上的任意非空开集都可以写成有限个或可数个互不相交的开区间的并集。

\vspace{0.2cm}

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\item  %Problem 07
什么是康托尔三分集？证明康托尔三分集是一个完备集、疏朗集、测度为零、但是基数仍为 $\aleph$. 

\vspace{0.2cm}

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\item  %Problem 08
计算 Cantor 三分集、Koch 曲线和 Sierpinski 地毯的 Hausdorff 维数。

\vspace{0.2cm}

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\item  %Problem 09
证明：用10进位小数表示 $[0,1]$ 中的数时，其用不着数字7的一切数组成一个完备集。

\vspace{0.2cm}

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\item  %Problem 10
证明：函数 $f(x)$ 为 $[a,b]$ 上的连续函数的充分必要条件是对任意实数 $c$, 集合
$E_1=\{x: f(x)\ge c\}$ 与 $E_2=\{x: f(x)\le c\}$ 都是闭集。

\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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